Theorems · Definition · algebraic geometry

AlgebraicGeometry.Scheme.Cover.ColimitGluingData.glued

{P : CategoryTheory.MorphismProperty AlgebraicGeometry.Scheme} →
  [inst : P.IsStableUnderBaseChange] →
    [P.IsMultiplicative] →
      {S : AlgebraicGeometry.Scheme} →
        {J : Type u_1} →
          [inst_2 : CategoryTheory.Category.{v_1, u_1} J] →
            {D : CategoryTheory.Functor J (P.Over ⊤ S)} →
              {𝒰 : S.OpenCover} →
                [inst_3 : CategoryTheory.Category.{v_2, u_2} 𝒰.I₀] →
                  [inst_4 : AlgebraicGeometry.Scheme.Cover.LocallyDirected 𝒰] →
                    AlgebraicGeometry.Scheme.Cover.ColimitGluingData D 𝒰 →
                      [∀ {i j : 𝒰.I₀} (hij : i ⟶ j),
                            CategoryTheory.Limits.PreservesColimitsOfShape J
                              (CategoryTheory.MorphismProperty.Over.pullback P ⊤
                                (AlgebraicGeometry.Scheme.Cover.trans 𝒰 hij))] →
                        [Quiver.IsThin 𝒰.I₀] →
                          [Small.{u, u_2} 𝒰.I₀] → [AlgebraicGeometry.IsZariskiLocalAtTarget P] → P.Over ⊤ S

The result of gluing the colim Dᵢ.

Defined in

Mathlib.AlgebraicGeometry.ColimitsOver

Cited by

7 results in Mathlib

Foundations

Depth 205 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.MorphismProperty.IsStableUnderBaseChangeCategoryTheory.MorphismProperty.IsMultiplicativeCategoryTheory.CategoryCategoryTheory.CategoryAlgebraicGeometry.Scheme.Cover.LocallyDirectedCategoryTheory.Limits.PreservesColimitsOfShapeQuiver.IsThinSmallAlgebraicGeometry.IsZariskiLocalAtTarget